In the case that the path cost is zero \(V({\bf x})=0\) , the artificially and naturally selected trajectories become distinct only due to the contributions from the end-point cost at \(t=t_f\) . For the choice of a linear evolutionary force \(A({\bf x}) = -2 K C {\bf x}\)  and a quadratic end-point cost, \(Q({\bf x},{t_f})= \frac{1}{2} ({\bf x}_{t_f}-{\bf x}^*)^\top \tilde{G} ({\bf x}_{t_f}-{\bf x}^*)\) , evolution follows an Ornstein-Uhlenbeck (OU) process and the solution to eq. (REF ) takes a Gaussian form (see e.g. ref. [1]}),

\(\nonumber P_u({\bf x},t) &=& \int d{\bf x}_{t_f} P_0 ({\bf x}_{t_f},t_f | {\bf x},t) P_0 ({\bf x},t) \exp \left[- Q({\bf x}_{t_f})/\lambda \right] \\&\sim & \int d{\bf x}_{t_f} \exp \left[ \frac{-1}{2}\left({\bf x}_{t_f}- \mu ({\bf x},\tau )\right)^\top K^{-1}_\tau \left({\bf x}_{t_f}- \mu ({\bf x},\tau )\right)\right]P_0 ({\bf x},t) \exp \left[- Q({\bf x}_{t_f})/\lambda \right]\) 
